Just a nitpick... it's a traffic lane. Motorcycles, trucks, cars, buses, and bicycles are all permitted to use it. But calling it a "car lane" bugs me as much as people saying "ATM Machine".

FWIW, if a majority of traffic may be turning right on red, I move to the far left of the lane I'm in (assuming it's a straight / right turn lane). That allows cars to still turn right on red.

If it's a straight only lane, I sit right in the middle of the lane until I clear the intersection. I have had too many unsafe passes (on both the left and the right) with other lane positions.

All that assumes that I'm in front of the cars. Usually, I'm sitting in the lane behind some other vehicles. But my center lane position is still used until I clear the intersection.